Geometry of Linear Algebra

MIT 18.06SC Linear Algebra, Fall 2011
Instructor: Linan Chen

A teaching assistant works through a problem on the geometry of linear algebra.

License: Creative Commons BY-NC-SA

[kmunson]

I'm not going to say this is the only person I can learn linear algebra from. But after looking at 100's of videos from dozens of creators/providers, I get the topic faster from Linan Chen than anyone else I have come across. While the content (the curriculum) seems very similar to others, the slight but meaningful differences in the way it is presented make a lot of difference. The cadence, the choice of words, its hard to put my finger on the exact reason but clearly Linan has a gift for teaching.

[handsomewang]

These are the straightest hand written lines I've ever seen.

[Axle_Max]

I have been looking for over a week for an explanation of why polynomial linear regression uses A -1 * A and this video explains it unbelievably simply!! Amazing. One of the biggest "lightbulb moments" of my life. Thank you so much.

[daudcodeslive]

Wow. Could she do this for every video? Amazing delivery. Kudos!

Linan has done such a good job on the explanation! Thanks, MIT OCW

[blankvoidsea]

Since the column picture visualizes the coefficients vectors, the axis labels for the column picture should not be x and y.
x and y - although variables - were impersonated as the 'dummy coefficients' for those vectors / linear combination.

[Deristrome]

This is from the year I was born in. Now I’m self studying linear algebra. Currently at differential equations. I hope I make it into MIT for transfer. Hard as it may be.

[toutane]

You really feel that every word has been weighed up to make it as clear, precise and natural as possible.

[ciai]

No wonder why MIT is recognized as one of the best schiool in the US. The students are so fortunate to have this quality of lecturers. I have study linear algebra with different teacher, and they were enable to make such a great review. THANK YOU MIT

[hongz]

Excellent teaching! More videos from her please.

[mrvzhao]

Don't get me wrong, I think this recitation is expertly done, no question about that. However, I'm a bit disappointed that some of the more important insights from this row/column picture comparison wasn't made clearer to viewers. For example…

In the ROW picture, you're looking at variables X and Y as two dimensions of the plane. The lines (as dictated by the linear equations) depict two constraints that the solution (a combination of X and Y, i.e. a point on the plane) must simultaneously satisfy. You obviously find the solution by reading off the X, Y coordinates of the intersection point. More generally, there can be many, many constraints on a lot of variables, but each ROW only describes ONE constraint on ALL variables.

Whereas in the column picture, you're NOT looking at the X-Y plane anymore (as someone already pointed out). The number of dimensions of the space you’re looking at reflects the number of constraints imposed by the linear system you’re dealing with. Roughly speaking, the more equations (or more rows) you have, the more constraints there are, the higher dimensional the space is. A column (shown as a vector in the space) describes a transformation in that space to be scaled by a variable. Each variable has one transformation (one vector) associated with it. Then the system is simply a linear combination of those transformations, constrained by the condition that the end result must equal to the column vector on the right side of the equal sign – which itself is a transformation. You can think of a COLUMN as ALL the constraints on ONE variable. So, to solve the equations, you need to find combinations of variables that will collectively scale/combine all the transformations just the right way, so that the total effect is the same as the right-hand-side transformation.

But to be fair, a lot of this should have been introduced in the lectures, not as part of a recitation.

[algotrader]

Looked her up to see what kind of Prof she has become, doesn't disappoint. Very highly rated professor.

[randomperson-wpwf]

Best and clearest video covering vector and matrix for linear algebra

[aakashyadav]

man!! the writing is so beautiful... when i tried to write on board in school it sucks...

explanation is way easy to go through...

[실버벨-fi]

I love the expression of v1x+v2y instead of xv1+yv2. Because it seems easier to understand for beginner. (although scalar *vector is more common.) But last part regarding inverse matrix seems too difficult topic in this stage.

[jun]

She speaks in a mind-easing tempo, like a flying assistant does.

[tonyavito]

Hi, I think in the column picture, it is advisable not to use the coordinates of the graph as x and y... this might confuse a student.
Besides that, every thing is just great!
Thank you!

[실버벨-fi]

I agree that the x and y axes for the column picture are confusing. But, thinking for the other axes and their unit vector is very interesting for me. I feel I am in totally different space!

[runningcat]

Just a suggestion: You have used x, y to stand for coefficients, then using xOy as vector coordinates could be confusing.

[mkonnaris]

Wow. This is exactly what I needed. Thank you